Variational superposed Gaussian approximation for time-dependent solutions of Langevin equations

TL;DR

This paper introduces a variational superposed Gaussian approximation (VSGA) method to solve time-dependent Langevin equations, accurately capturing probability densities and moments under complex signals and noise types.

Contribution

The paper presents a novel VSGA method for dynamical Langevin equations, effective for chaotic signals and incorporating white and colored noise, surpassing traditional Fourier approaches.

Findings

VSGA accurately matches Monte Carlo simulations for PDFs.

The method reveals stochastic resonance phenomena under various noise conditions.

VSGA effectively handles non-periodic, chaotic input signals.

Abstract

We propose a variational superposed Gaussian approximation (VSGA) for dynamical solutions of Langevin equations subject to applied signals, determining time-dependent parameters of superposed Gaussian distributions by the variational principle. We apply the proposed VSGA to systems driven by a chaotic signal, where the conventional Fourier method cannot be adopted, and calculate the time evolution of probability density functions (PDFs) and moments. Both white and colored Gaussian noises terms are included to describe fluctuations. Our calculations show that time-dependent PDFs obtained by VSGA agree excellently with those obtained by Monte Carlo simulations. The correlation between the chaotic input signal and the mean response are also calculated as a function of the noise intensity, which confirms the occurrence of aperiodic stochastic resonance with both white and colored noises.